jee-main

Papers (169)
2025
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2024
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2023
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2022
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2021
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2020
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2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 30 session1_10jan_shift2 12 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 20 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 3 session2_12apr_shift1 3 session2_12apr_shift2 9
2018
08apr 29 15apr 28 15apr_shift1 28 15apr_shift2 2 16apr 15
2017
02apr 28 08apr 29 09apr 30
2016
03apr 30 09apr 30 10apr 28
2015
04apr 29 10apr 30
2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
2013
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
2012
07may 18 12may 22 19may 13 26may 17 offline 30
2011
jee-main_2011.pdf 13
2010
jee-main_2010.pdf 1
2009
jee-main_2009.pdf 1
2008
jee-main_2008.pdf 1
2007
jee-main_2007.pdf 38
2005
jee-main_2005.pdf 19
2004
jee-main_2004.pdf 11
2003
jee-main_2003.pdf 9
2002
jee-main_2002.pdf 8
2018 15apr

28 maths questions

Q61 Solving quadratics and applications Optimization or extremal value of an expression via completing the square View
If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x ^ { 2 } + ( 2 - \lambda ) x + ( 10 - \lambda ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is :
(1) $4 \sqrt { 2 }$
(2) 20
(3) $2 \sqrt { 5 }$
(4) $2 \sqrt { 7 }$
Q62 Standard trigonometric equations Evaluate trigonometric expression given a constraint View
If $\tan A$ and $\tan B$ are the roots of the quadratic equation $3 x ^ { 2 } - 10 x - 25 = 0$, then the value of $3 \sin ^ { 2 } ( A + B ) - 10 \sin ( A + B ) \cos ( A + B ) - 25 \cos ^ { 2 } ( A + B )$ is :
(1) - 25
(2) 10
(3) - 10
(4) 25
Q63 Complex Numbers Arithmetic Identifying Real/Imaginary Parts or Components View
The set of all $\alpha \in R$, for which $w = \frac { 1 + ( 1 - 8 \alpha ) z } { 1 - z }$ is a purely imaginary number, for all $z \in C$ satisfying $| z | = 1$ and $\operatorname { Re } ( z ) \neq 1$, is :
(1) $\{ 0 \}$
(2) $\left\{ 0 , \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right\}$
(3) equal to $R$
(4) an empty set
Q64 Permutations & Arrangements Forming Numbers with Digit Constraints View
$n$-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of $n$ for which 900 such distinct numbers can be formed is :
(1) 9
(2) 7
(3) 8
(4) 6
Q65 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
If $b$ is the first term of an infinite geometric progression whose sum is five, then $b$ lies in the interval
(1) $[ 10 , \infty )$
(2) $( - \infty , - 10 ]$
(3) $( - 10,0 )$
(4) $( 0,10 )$
Q66 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
If $x _ { 1 } , x _ { 2 } , \ldots\ldots , x _ { n }$ and $\frac { 1 } { h _ { 1 } } , \frac { 1 } { h _ { 2 } } , \ldots\ldots , \frac { 1 } { h _ { n } }$ are two A.P.s such that $x _ { 3 } = h _ { 2 } = 8 \& x _ { 8 } = h _ { 7 } = 20$, then $x _ { 5 } \cdot h _ { 10 }$ is equal to
(1) 3200
(2) 1600
(3) 2650
(4) 2560
Q67 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
If $n$ is the degree of the polynomial, $\left[ \frac { 2 } { \sqrt { 5 x ^ { 3 } + 1 } - \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 } + \left[ \frac { 2 } { \sqrt { 5 x ^ { 3 } + 1 } + \sqrt { 5 x ^ { 3 } - 1 } } \right] ^ { 8 }$ and $m$ is the coefficient of $x ^ { n }$ in it, then the ordered pair $( n , m )$ is equal to
(1) $\left( 8,5 ( 10 ) ^ { 4 } \right)$
(2) $\left( 12,8 ( 10 ) ^ { 4 } \right)$
(3) $\left( 12 , ( 20 ) ^ { 4 } \right)$
(4) $\left( 24 , ( 10 ) ^ { 8 } \right)$
Q68 Circles Circle Equation Derivation View
A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line $y - 4 x + 3 = 0$, then its radius is equal to :
(1) $\sqrt { 5 }$
(2) $\sqrt { 2 }$
(3) 2
(4) 1
Q69 Circles Tangent Lines and Tangent Lengths View
Two parabolas with a common vertex and with axes along the $x$-axis and $y$-axis respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is :
(1) $3 ( x + y ) + 4 = 0$
(2) $8 ( 2 x + y ) + 3 = 0$
(3) $x + 2 y + 3 = 0$
(4) $4 ( x + y ) + 3 = 0$
Q70 Tangents, normals and gradients Normal or perpendicular line problems View
If $\beta$ is one of the angles between the normals to the ellipse $x ^ { 2 } + 3 y ^ { 2 } = 9$ at the points $( 3 \cos \theta , \sqrt { 3 } \sin \theta )$ and $( - 3 \sin \theta , \sqrt { 3 } \cos \theta ) ; \theta \in \left( 0 , \frac { \pi } { 2 } \right) ;$ then $\frac { 2 \cot \beta } { \sin 2 \theta }$ is equal to :
(1) $\frac { 1 } { \sqrt { 3 } }$
(2) $\frac { \sqrt { 3 } } { 4 }$
(3) $\frac { 2 } { \sqrt { 3 } }$
(4) $\sqrt { 2 }$
Q71 Circles Circle-Related Locus Problems View
If the tangent drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinates axes at the distinct points $A$ and $B$, then the locus of the midpoint of $AB$ is :
(1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(3) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(4) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
Q73 Measures of Location and Spread View
The mean of a set of 30 observation is 75. If each observations is multiplied by non-zero number $\lambda$ and then each of them is decreased by 25, their mean remains the same. Then, $\lambda$ is equal to :
(1) $\frac { 4 } { 3 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 10 } { 3 }$
(4) $\frac { 2 } { 3 }$
Q74 Projectiles Horizontal Launch or Dropped Object Problems View
An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt { 3 } \mathrm {~km}$ above it is observed at an elevation of $60 ^ { \circ }$ from a point on the ground. If after five seconds, its elevation from the same point is $30 ^ { \circ }$, then the speed (in $\mathrm { km } / \mathrm { hr }$ ) of the aeroplane is
(1) 720
(2) 1500
(3) 750
(4) 1440
Q75 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
In a triangle $ABC$, coordinates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are respectively, $x + y = 5$ and $x = 4$. Then area of $\triangle ABC$ (in sq. units) is :
(1) 12
(2) 4
(3) 9
(4) 5
Q77 Matrices Matrix Algebra and Product Properties View
Let $A$ be a matrix such that $A \cdot \left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$ is a scalar matrix and $| 3 A | = 108$. Then, $A ^ { 2 }$ equals :
(1) $\left[ \begin{array} { c c } 4 & 0 \\ - 32 & 36 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 36 & - 32 \\ 0 & 4 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 36 & 0 \\ - 32 & 4 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 4 & - 32 \\ 0 & 36 \end{array} \right]$
Q78 Chain Rule Limit Involving Derivative Definition of Composed Functions View
If $f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$
(1) does not exist
(2) exists and is equal to $-2$
(3) exists and is equal to 0
(4) exists and is equal to 2
Q79 Simultaneous equations View
Let $S$ be the set of all real values of $k$ for which the system of linear equations $x + y + z = 2$ $2 x + y - z = 3$ $3 x + 2 y + k z = 4$ has a unique solution. Then, $S$ is :
(1) equal to $R - \{ 0 \}$
(2) an empty set
(3) equal to $R$
(4) equal to $\{ 0 \}$
Q80 Differential equations Qualitative Analysis of DE Solutions View
Let $S = \left\{ ( \lambda , \mu ) \in R \times R : f ( t ) = \left( | \lambda | e ^ { | t | } - \mu \right) \sin ( 2 | t | ) , t \in R \right.$ is a differentiable function $\}$. Then, $S$ is a subset of :
(1) $( - \infty , 0 ) \times R$
(2) $R \times [ 0 , \infty )$
(3) $[ 0 , \infty ) \times R$
(4) $R \times ( - \infty , 0 )$
Q81 Implicit equations and differentiation Second derivative via implicit differentiation View
If $x ^ { 2 } + y ^ { 2 } + \sin y = 4$, then the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( - 2,0 )$ is :
(1) $-34$
(2) 4
(3) $-2$
(4) $-32$
Q82 Applied differentiation Applied modeling with differentiation View
If a right circular cone, having maximum volume, is inscribed in a sphere of radius $3$ cm, then the curved surface area (in $\mathrm { cm } ^ { 2 }$) of this cone is :
(1) $8 \sqrt { 2 } \pi$
(2) $6 \sqrt { 2 } \pi$
(3) $8 \sqrt { 3 } \pi$
(4) $6 \sqrt { 3 } \pi$
Q83 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
If $f \left( \frac { x - 4 } { x + 2 } \right) = 2 x + 1 , ( x \in R - \{ 1 , - 2 \} )$, then $\int f ( x ) d x$ is equal to
(1) $12 \ln | 1 - x | - 3 x + C$
(2) $- 12 \ln | 1 - x | - 3 x + C$
(3) $12 \ln | 1 - x | + 3 x + C$
(4) $- 12 \ln | 1 - x | + 3 x + C$
Q84 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin ^ { 4 } x \left( 1 + \ln \left( \frac { 2 + \sin x } { 2 - \sin x } \right) \right) d x$ is
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 8 } \pi$
(3) 0
(4) $\frac { 3 } { 16 } \pi$
Q85 Areas by integration View
The area (in sq. units) of the region $\{ x \in R : x \geq 0 , y \geq 0 , y \geq x - 2$ and $y \leq \sqrt { x } \}$ is
(1) $\frac { 13 } { 3 }$
(2) $\frac { 8 } { 3 }$
(3) $\frac { 10 } { 3 }$
(4) $\frac { 5 } { 3 }$
Q86 First order differential equations (integrating factor) View
Let $y = y ( x )$ be the solution of the differential equation $\frac { d y } { d x } + 2 y = f ( x )$, where $f ( x ) = \left\{ \begin{array} { l l } 1 , & x \in [ 0,1 ] \\ 0 , & \text { otherwise } \end{array} \right.$. If $y ( 0 ) = 0$, then $y \left( \frac { 3 } { 2 } \right)$ is
(1) $\frac { e ^ { 2 } - 1 } { e ^ { 3 } }$
(2) $\frac { 1 } { 2 e }$
(3) $\frac { e ^ { 2 } + 1 } { 2 e ^ { 4 } }$
(4) $\frac { e ^ { 2 } - 1 } { 2 e ^ { 3 } }$
Q87 Vectors Introduction & 2D Dot Product Computation View
If $\vec { a } , \vec { b } , \vec { c }$ are unit vectors such that $\vec { a } + 2 \vec { b } + 2 \vec { c } = \overrightarrow { 0 }$, then $| \vec { a } \times \vec { c } |$ is equal to :
(1) $\frac { 1 } { 4 }$
(2) $\frac { 15 } { 16 }$
(3) $\frac { \sqrt { 15 } } { 4 }$
(4) $\frac { \sqrt { 15 } } { 16 }$
Q88 Vectors: Lines & Planes Coplanarity and Relative Position of Planes View
A variable plane passes through a fixed point $( 3,2,1 )$ and meets $x , y$ and $z$-axes at $A , B \& C$ respectively. A plane is drawn parallel to the $yz$-plane through $A$, a second plane is drawn parallel to the $zx$-plane through $B$ and a third plane is drawn parallel to the $xy$-plane through $C$. Then the locus of the point of intersection of these three planes, is
(1) $\frac { 3 } { x } + \frac { 2 } { y } + \frac { 1 } { z } = 1$
(2) $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = \frac { 11 } { 6 }$
(3) $x + y + z = 6$
(4) $\frac { x } { 3 } + \frac { y } { 2 } + \frac { z } { 1 } = 1$
Q89 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
An angle between the plane $x + y + z = 5$ and the line of intersection of the planes, $3 x + 4 y + z - 1 = 0$ and $5 x + 8 y + 2 z + 14 = 0$ is
(1) $\cos ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$
(2) $\cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(4) $\sin ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$
Q90 Conditional Probability Bayes' Theorem with Production/Source Identification View
A box $A$ contains 2 white, 3 red and 2 black balls. Another box $B$ contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box $B$ is :
(1) $\frac { 7 } { 8 }$
(2) $\frac { 9 } { 16 }$
(3) $\frac { 7 } { 16 }$
(4) $\frac { 9 } { 32 }$